x y z dr where dr d x\u02c6 \u0131 dy \u02c6 dz \u02c6 kd For any differential form \u03c9 d d \u03c9

# X y z dr where dr d xˆ ı dy ˆ dz ˆ kd for any

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( x , y , z ) ¨ d r where d r = d x ˆ ı + d y ˆ + d z ˆ k (d) For any differential form ω , d ( d ω ) = 0 Definition 4.7.11 (Differentiation of differential forms) . Example 4.7.12 (a) If f ( x , y , z ) = x , then d f = B x B x ( x , y , z ) d x + B x B y ( x , y , z ) d y + B x B z ( x , y , z ) d z = d x 63 That d is unique just means that the action of d on any differential form is completely determined by the four rules (a), (b), (c), (d). We will see in Example 4.7.12 .b,c,d, that this is indeed the case. 241
I NTEGRAL T HEOREMS 4.7 O PTIONAL —- A G ENERALIZED S TOKES ’ T HEOREM That is, d x really is the operator d applied to the function x . Similarly, d y really is the operator d applied to the function y and d z really is the operator d applied to the function z . (b) For any k –form ω d ω ^ d x = d ω ^ d x + ( ´ 1 ) k ω ^ d ( d x ) = d ω ^ d x Similarly d ω ^ d y = d ω ^ d y d ω ^ d z = d ω ^ d z (c) For any 1–form d F 1 d x + F 2 d y + F 3 d z = d F 1 ^ d x + d F 2 ^ d y + d F 3 ^ d z = B F 1 B x d x + B F 1 B y d y + B F 1 B z d z ^ d x + B F 2 B x d x + B F 2 B y d y + B F 2 B z d z ^ d y + B F 3 B x d x + B F 3 B y d y + B F 3 B z d z ^ d z = B F 3 B y ´ B F 2 B z d y ^ d z + B F 1 B z ´ B F 3 B x d z ^ d x + B F 2 B x ´ B F 1 B y d x ^ d y = ( ˆ F ) 1 d y ^ d z + ( ˆ F ) 2 d z ^ d x + ( ˆ F ) 3 d x ^ d y (d) For any 2–form d F 1 d y ^ d z + F 2 d z ^ d x + F 3 d x ^ d y = d F 1 ^ d y ^ d z + d F 2 ^ d z ^ d x + d F 3 ^ d x ^ d y = B F 1 B x d x + B F 1 B y d y + B F 1 B z d z ^ d y ^ d z + B F 2 B x d x + B F 2 B y d y + B F 2 B z d z ^ d z ^ d x + B F 3 B x d x + B F 3 B y d y + B F 3 B z d z ^ d x ^ d y = B F 1 B x + B F 2 B y + B F 3 B z d x ^ d y ^ d z = ¨ F d x ^ d y ^ d z (e) For any 3–form d f d x ^ d y ^ d z = B f B x d x + B f B y d y + B f B z d z ^ d x ^ d y ^ d z = 0 242
I NTEGRAL T HEOREMS 4.7 O PTIONAL —- A G ENERALIZED S TOKES ’ T HEOREM Example 4.7.12 Example 4.7.13 In Definition 4.7.11 .c, we defined, for any function f ( x , y , z ) of three variables d f = B f B x ( x , y , z ) d x + B f B y ( x , y , z ) d y + B f B z ( x , y , z ) d z The analogous formulae 64 for functions of one or two variables also apply. d f ( t ) = d f d t ( t ) d t d f ( uv ) = B f B u ( u , v ) d u + B f B v ( u , v ) d v (a) Let F 1 ( x , y , z ) d x + F 2 ( x , y , z ) d y + F 3 ( x , y , z ) d z be a 1–form. Suppose that we substi- tute x = x ( t ) , y = y ( t ) and z = z ( t ) , so that we are restricting our 1–form to a parametrized curve. Then, writing r ( t ) = ( x ( t ) , y ( t ) , z ( t ) ) , F 1 ( x ( t ) , y ( t ) , z ( t ) ) d x ( t ) + F 2 ( x ( t ) , y ( t ) , z ( t ) ) d y ( t ) + F 3 ( x ( t ) , y ( t ) , z ( t ) ) d z ( t ) = F 1 ( r ( t ) ) d x d t ( t ) d t + F 2 ( r ( t ) ) d y d t ( t ) d t + F 3 ( r ( t ) ) d z d t ( t ) d t = F ( r ( t ) ) ¨ d r d t ( t ) d t (b) Let F 1 ( x , y , z ) d y ^ d z + F 2 ( x , y , z ) d z ^ d x + F 3 ( x , y , z ) d x ^ d y be a 2–form. Suppose that we substitute x = x ( u , v ) , y = y ( u , v ) and z = z ( u , v ) , so that we are restricting our 2–form to a parametrized surface. Then, writing r ( u , v ) = ( x ( u , v ) , y ( u , v ) , z ( u , v ) ) , F 1 ( x ( u , v ) , y ( u , v ) , z ( u , v ) ) d y ( u , v ) ^ d z ( u , v ) + F 2 ( x ( u , v ) , y ( u , v ) , z ( u , v ) ) d z ( u , v ) ^ d x ( u , v ) + F 3 ( x ( u , v ) , y ( u , v ) , z ( u , v ) ) d x ( u , v ) ^ d y ( u , v ) = F 1 ( r ( u , v ) ) B y B u d u + B y B v d v ^ B z B u d u + B z B v d v + F 2 ( r ( u , v ) ) B z B u d u + B z B v d v ^ B x B u d u + B x B v d v + F 3 ( r ( u , v ) ) B x B u d u + B x B v d v ^ B y B u d u + B y B v d v = h F 1 ( r ( u , v ) ) B y B u B z B v ´ B y B v B z B u + F 2 ( r ( u , v ) ) B z B u B x B v ´ B z B v B x B u + F 3 ( r ( u , v ) ) B x B u B y B v ´ B x B v B y B u i d u ^ d v = h F ( r ( u , v ) ) ¨ B r B u ( u , v ) ˆ B r B v ( u , v ) i d u ^ d v 64 Indeed, you can view f ( t ) as a function of three variables that happens to be independent of two of the three variables. Similarly you can view f ( u , v )